Design of Large Effective Apertures for Millimeter Wave Systems Using A

IEEE TRANSACTIONS ON SIGNAL PROCESSING 1

Design of Large Effective Apertures for Millimeter Wave Systems using a Sparse Array of Subarrays Anant Gupta, Student Member, IEEE, Upamanyu Madhow, Fellow, IEEE Amin Arbabian, Senior Member, IEEE and Ali Sadri

Abstract—We investigate synthesis of a large effective aperture relatively compact apertures. As a running example throughout using a sparse array of subarrays. We employ a multi-objective this paper, we consider the design of a 60 GHz array of optimization framework for placement of subarrays within a subarrays created by placing 8 subarrays over an aperture size prescribed area dictated by form factor constraints, trading off the smaller beam width obtained by spacing out the subarrays of 10 cm by 10 cm (20λ × 20λ for λ = 0.5 cm), where each against the grating and side lobes created by sparse placement. subarray has 4 × 4 elements arranged in uniform rectangular We assess the performance of our designs for the fundamental grid with 0.5λ horizontal spacing and 0.6λ vertical spacing. problem of bearing estimation for one or more sources, compar- The subarrays need to be aligned along their axes, assuming ing performance against estimation-theoretic bounds. Our tiled that all elements have unidirectional linear polarization. Each architecture is motivated by recent progress in low-cost hardware realizations of moderately sized antenna arrays (which play the subarray tile occupies extra physical area on the plane, which role of subarrays) in the millimeter wave band, and our numerical must also be accounted for in the placement procedure. examples are based on 16-element (4 × 4) subarrays in the 60 These design restrictions are consistent with existing prototype GHz unlicensed band. subarrays. Index Terms—Millimeter wave radar, Estimation Bounds, Compressive Estimation, Gridless Super-Resolution, Sparse Sub- A. Contributions array design, Multi-objective Optimization. Our contributions are summarized as follows: • We formulate the problem of subarray placement as multi- I.INTRODUCTION objective optimization of key performance measures such Many sensing and situational awareness applications (e.g., as beam width, maximum sidelobe level, eccentricity and radar imaging for vehicles and drones) require highly direc- directivity, tional, electronically steerable beams. Reducing beam width Minimize BW(C, w), MSLL(C, w), ecc(C, w) requires expansion of antenna aperture. This is typically accomplished by filling the aperture with antenna elements Maximize GD(C, w) (1) spaced at half the carrier wavelength or less, in order to avoid subject to AOS(C) grating lobes. However, this approach does not scale well with where C is N × 2 Subarray center position matrix, w is aperture size since the cost, power consumption and design s N × 1 beamsteering weight vector and AOS(C) are physical complexity increases with number of antenna elements. constraints to avoid overlapping subarrays. Note that orien- In this paper, we investigate the problem of synthesizing tation of subarrays is not an optimization variable in this narrow beams using a tiled architecture, with a sparse set of architecture, since the polarization of the elements has to subarrays spread over a large physical aperture. Each subarray be aligned for beamforming. The problem (1) is non-convex is a relatively compact antenna array with a moderate number and combinatorially explosive. We use geometric heuristics of elements. The total number of antenna elements, summing to eliminate similar configurations in the first stage of our over subarrays, is much smaller than that for a classical algorithm, and then employ a second stage of refinement using arXiv:1907.10749v1 [eess.SP] 24 Jul 2019 design spanning the entire physical aperture. The sidelobes small perturbations around the first stage solution. and grating lobes resulting from such spatial undersampling • We evaluate our designs using estimation-theoretic bench- must therefore be controlled in order for our proposed “array marks for two-dimensional (2D) direction of arrival (DoA) of subarrays” to be useful. Our goal here is to determine the estimation. At low signal-to-noise ratio (SNR), large sidelobes placement of a given number of subarrays over a physical aper- can lead to large errors in the DoA estimate. At high SNR, on ture in order to optimize multiple beam attributes, including the other hand, the DoA estimation error is governed by beam beam width, maximum sidelobe level, and directivity. width. We derive a Ziv-Zakai bound (ZZB), which captures While our framework is general, the design of millimeter the effect of both large and small estimation errors, for DoA wave arrays is of particular interest to us, because the small estimation for specular paths. The ZZB exhibits a distinct carrier wavelength enables synthesis of narrow beams using transition in its behavior from low to high SNR, tending at A. Gupta and U. Madhow are with the Department of Electrical and high SNR to the Cramer-Rao bound (CRB), which captures the Computer Engineering, University of California Santa Barbara, CA USA effect of small errors around the true parameter value. Thus, (Email: {anantgupta, madhow}@ece.ucsb.edu) we use the ZZB transition SNR as a measure of efficacy of A. Arbabian is with the Department of Electrical Engineering, Stanford University, CA USA (Email: [email protected]) sidelobe reduction, and the CRB as a measure of efficacy of A. Sadri is with Intel Corporation (Email: [email protected] beam width reduction. IEEE TRANSACTIONS ON SIGNAL PROCESSING 2

• The beam attributes of arrays designed using multi- Denoising (BPDN) [14] and Newtonized Orthogonal Matching objective optimization depend on its parameterization. We Pursuit (NOMP) [15] are both more general and have better design two sparse arrays, A1 with primary emphasis on performance. It is worth noting that [16] shows that, for reducing beamwidth and A2 based on joint optimization of large arrays, BPDN and other sparse estimation techniques beamwidth and maximum sidelobe level. These designs are with compressive measurements outperform subspace-based compared against two benchmark arrays. The first is termed a methods. In this paper, we employ NOMP in our numer- “compact array,” with subarrays packed closely together: this ical experiments, since we have found it to provide better is expected to have worse beam width but smaller sidelobes performance than BPDN at lower complexity. We also show than our sparse designs. The second is termed a “naive that the performance trends are unchanged under compressive array,” obtained by placing subarrays in diamond pattern to measurements, consistent with recent general theory [17]. obtain beamwidth equivalent to that of sparse array A2. Some illustrative numerical results for our running example are as C. Outline follows. The sparse design A1 is 11 dB better than the compact array in terms of CRB, while degrading less than 1 dB in terms We first describe the beam attributes to be optimized while of ZZB threshold. The sparse design A2 is 4 dB better in terms designing the sparse arrays and discuss constraints for the of CRB than the “naive array,” while also having a better ZZB optimization in Section II. The geometric heuristics and design threshold. approach is described in detail in Section III. We then provide • We investigate DoA estimation performance numerically a brief review of estimation bounds for 2D bearing estimation using a state of the art super-resolution algorithm. The impact and discuss their utility for analysing the sparse arrays in Sec- of the higher sidelobes due to sparse placement, and hence tion IV. Finally, we compare the example array configurations that of our optimization procedure, is more evident when against benchmarks and evaluate the performance of super- estimating DoA in the presence of multiple interfering targets. resolution algorithms in Section V. We show that, depending on the strength of interferers, our optimized arrays achieve better estimation accuracy than the II.SPARSE SUBARRAY DESIGN “compact” and “naive” benchmark arrays at moderate to high We formulate the array design problem in terms of jointly SNR, due to a combination of sharper beamwidth and lower optimizing multiple beam parameters that are expected to sidelobes. We also show that the efficacy of DoA estimation affect DoA estimation performance. using our sparse designs, and the associated benchmarks, is maintained when we employ compressive measurements. (a) (b) z B. Related Work DoA Subarray There is a rich body of work on sparsifying linear arrays, including minimum redundancy arrays [1], genetic optimiza- Grid tion [2], joint Cramer´ Rao Bound and sidelobe level optimiza- v tion [3], and simulated annealing [4]. Most popular design y strategies try to find an element pattern which minimizes θmax beamwidth, along with some notion of DoA ambiguities Directional cosines such as sidelobe level or probability of DoA outlier. Recent u = sin(θ) cos(φ) v = sin(θ) sin(φ) approaches like Nested 2D arrays [5] and H-arrays [6] utilize Unit Circle the idea of “difference co-array” to reduce the number of x redundant spacings and maximize the randomness of element Fig. 1. (a) 2D Array Geometry and Spherical coordinate system. (b)ROI: positions, so that the number of spatial frequencies being Uniform distribution of 2D-DoA u in spherical cap with half angle θmax sampled by the array is maximized. Most of these techniques, however, assume that antenna elements can be placed freely. Hence, they do not apply in our setting, where element A. Beam Pattern Basics placement within subarrays is constrained. The prior work most similar to our is [7], which investigates We use the directional cosines design of linear arrays with two and three subarrays. However, u = sin(θ) cos(φ), v = sin(θ) sin(φ) (2) the focus there is on performance criteria for comparing a number of sensible designs in a far smaller design space, rather to represent the DoA of target. The elevation θ and azimuth than searching over a large space of possibilities as we do here. φ angles are measured from the broadside direction (per- Our performance evaluation requires implementation of pendicular to the baseline array plane). The 2D beampattern DoA estimation algorithms. Classical subspace-based algo- R(u, v) in direction (u, v) when the beam is steered towards rithms such as MUSIC [8] and ESPRIT [9], as well as their the broadside is given by extensions to arrays of subarrays such as [10]–[13], rely on N 2 regular array geometries for efficient computation. Recently 1 X jk udx+vdy R(u, v) = e ( i i ) (3) developed super-resolution algorithms such as Basis Pursuit N 2 i=1 IEEE TRANSACTIONS ON SIGNAL PROCESSING 3

x y T where N is the number of array elements; [di , di ] , di 2π are the 2D co-ordinates of arrays elements, and k = λ is the wavenumber. We assume isotropic antenna elements with ideal steering weights and far-field sources with normalized response. The term “subarray” refers to the subset of elements with uniform half-wavelength spacing, while “super-array” refers to the placement of these subarrays, which is described by the subarray centers. Since the elements in a subarray are fixed, the array element locations, D can be expressed in terms 1 1 of the subarray centers, C, as D = C⊗ Ne +De⊗ Ns , where De is the fixed 2×Ne matrix containing the subarray element coordinates with respect to its center, Ns is the number of subarrays, Ne is number of elements in individual subarrays, 1n is an n × 1 column vector of ones, and ⊗ denotes the Kronecker product. When beamforming in a general direction (u0, v0) (broad- Fig. 2. Beam Attributes from array Beampattern. side corresponds to (u0, v0) = (0, 0)), the beam pattern is simply given by • Eccentricity (ecc): is a measure of the asymmetry of

R(u0,v0)(u, v) = R(u − u0, v − v0) (4) the mainbeam. We add this additional parameter to suppress the trivial linear placement solution, ecc = For Direction of Arrival (DoA) estimation, the ideal beam q 2 should have small beamwidth with minimal sidelobes and high 1 − (BWmin/BWmax) directivity. We consider the following beam attributes, some For non-uniform planar arrays, none of these beam parameters of which depend on the steering direction (u0, v0), as key have a closed form expression, hence they must be computed performance metrics to be optimized: numerically. In our simulations, we compute these beam • 2D beamwidth (BW): Although the mainbeam of non- attributes using a beampattern over a 512 × 512 grid in UV uniform array has non-trivial shape in 2D, we approx- space as shown in Figure 2. imate it as an ellipse to define beamwidth. We evaluate the 2D beamwidth in terms of the 3-dB beamwidths B. Problem Formulation along the major and minor axes of this ellipse, de- In order to develop geometric heuristics for optimization, we noted by BWmax and BWmin, respectively. The mean first analyze the effect of increasing aperture width, (keeping squared error of DoA estimation depends on the sum the number of antenna elements fixed, for linear and planar ar- of these beamwidths (see Appendix A), hence we define rays, with uniform and subarray-based architectures as shown BWDoA = pBW2 + BW2 beamwidth as max min. in the rightmost section of Figure 3. In the plots of beam • MSLL Maximum sidelobe level ( ): is the relative level attributes in Figure 3, the dashed line represents the aperture of the strongest sidelobe in the beampattern with re- width when inter-element spacing equals a half-wavelength, spect to mainlobe. MSLL = 10 log (Rmax/Rsidelobe). when the uniform and subarray-based configurations match. Rmax,Rsidelobe are the largest and second largest mag- • The MSLL for subarrayed array increases much faster nitude local maxima’s of beampattern R(u, v) given by, than uniform configuration due to grating lobe appearing Rmax(u0, v0) = max Ru ,v (u, v) close to mainbeam. This attribute is sensitive to element u,v 0 0 distribution and behaves unpredictably for non-uniform =R (u , v ) = R(0, 0) u0,v0 0 0 arrays. ∗ ∗ max Rsidelobe = max Ru ,v (u , v ) • The 3dB Beamwidth (BW for planar array) for both u∗,v∗ 0 0 array types reduces congruently reaffirming that it is s.t. (u∗, v∗) 6= (u , v ), 0 0 inversely proportional to aperture width independent of ∗ ∗ ∗ ∗ Ru0,v0 (u , v ) ≥ Ru0,v0 (D(u , v )) the distribution of elements. ∗ ∗ ∗ ∗ • Directivity increases as we increase the inter-element where D(u , v ) = {(u, v): |u − u | < , |v − v | < } spacing, but only upto certain limit and then becomes denotes -neighborhood. Note that Rmax does not depend constant [19]. This generalizes well to planar arrays as on steering direction (u0, v0), but Rsidelobe might. shown in the Figure 3. As one can see the directiv- • Directivity (G ): The directivity is the ratio of mainlobe D Rmax ity for subarrayed configurations remains approximately power to average power, GD = 10 log The av- Ravg constant with increasing aperture width beyond standard erage power does not have closed form expression for spacing. Hence we do not include this metric in our cost general planar arrays and is evaluated in (u, v) domain function. by the integral [18], √ The objectives that we wish to trade off against each other Z 1 Z 1−v2 do not have the same units: for example, MSLL is measured in 2 Ru0,v0 (u, v) Ravg = √ dudv √ 2 2 dB from max sidelobe level, whereas BW is measured in deg. 4π −1 − 1−v2 1 − u − v IEEE TRANSACTIONS ON SIGNAL PROCESSING 4

MSLL Beamwidth Directivity 0 30 30 Linear Subarray

-5 25 25 Linear Uniform

-10 20 Planar Subarray 20

-15 15 (dB) D

G 15 BW (deg) MSLL-20 (dB) 10

10 -25 5 Planar Uniform -30 0 5 2 4 6 8 10 2 4 6 8 10 2 4 6 8 10 Aperture width (λ) Aperture width (λ) Aperture width (λ) Linear Subarray Linear Uniform Planar Subarray Planar Uniform

Fig. 3. Comparison of beam attributes of subarrayed and uniform architecture with increasing aperture width.

We therefore normalize each raw objective value, oraw by its C. Invariance to Beamforming Direction range as follows: The cost function in (6) is evaluated using the beampattern R(u−u , v−v ), which depends on the beamsteering direction oraw(C) − min {oraw(C)} 0 0 ∀C (u , v ). It would be prohibitively expensive to evaluate the o(C) = 0 0 max {oraw(C)} − min {oraw(C)} beam attributes over all such beampatterns for finding the op- ∀C ∀C timal (C, w). However, for arrays with isotropic elements, we The range of each objective is computed numerically while can define an Expanded Beam pattern (EBP) which subsumes constructing the dictionary of all configurations, C ∈ C, beampatterns of all steering direction in a Region of Interest described later in III-A. (ROI) [20]. Suppose that our maximum steering angle in the The constrained multi-objective optimization can now be ROI is θmax. From (2), we see that (u0, v0) lies within a circle formulated as follows: of radius sin θmax. On the other hand, sidelobes can appear at any (u, v) within a circle of radius 1. It is easy to see, ∗ ∗ C , w =arg min f(C, w) therefore, that (u − u0, v − v0) is guaranteed to lie within a ∀C,w (5) circle of radius 1 + sin θmax. We can therefore compute beam subject to AOS(C) attributes using the following EBP: where N 2 1 X jkρ ud˜ x+˜vdy R (˜u, v˜) = e ( i i ) ρ N 2 (7) f(C, w) = αBW(C, w) + βMSLL(C, w) + γecc(C, w) (6) i=1 ρ = 1 + sin(θmax) is the weighted cost function in terms of the normalized Figure 4 shows the EBP, R1.5(˜u, v˜) for ROI with θmax = objective functions and [α, β, γ] are weights that can be used ◦ 30 , and the beampattern for the steering angle ((u0, v0) = to sweep through the optimal surface for this optimization. We (0.3, 0.4)). The shape of the main beam is preserved under show some example arrays obtained for different choices of the transformation (7), hence beam width and eccentricity can weights in Table I. be directly evaluated from (7). The MSLL evaluated from EBP This is a combinatorial optimization problem which is non- is a worst-case value, corresponding to an argument ρ(˜u, v˜) convex, with non-differentiable geometric constraints. The which, in principle, might not correspond to a feasible value of objective function can be evaluated exactly given the subarray (u−u0, v−v0) in (4). However, the maximum sidelobe always centers, but exploring the entire solution space is computation- lies within the main lobe of the subarray beam pattern, so that ally infeasible. Furthermore, beam characteristics in general physically implausible values of ρ(˜u, v˜) do not correspond to depend on the steering weights, which in turn depend on large local maxima of the EBP. the direction (u0, v0) in which we are steering. We therefore With the introduction of the EBP, we can, without loss of employ two key simplifications: generality, assume that the main beam is being steered towards • We remove the dependence of the cost function on broadside, setting w = 1N . Our problem now reduces to beamforming direction, and hence on steering weights, finding the optimal configuration C∗. by computing the objectives based on an expanded beam pattern, as discussed in Section II-C. III.PLACEMENT OPTIMIZATION • We employ geometric heuristics to cut down the solution In order to optimize the placement, we need to evaluate space to a reasonable size, as described in Section III. the cost function over all array configurations. The number IEEE TRANSACTIONS ON SIGNAL PROCESSING 5

that of the full array in the Dictionary search algorithm. Each node in the tree stores a subarray center position, and the path from root to a node at the nth layer of prefix tree represents a unique configuration of n subarrays. The subarray centers are constrained to lie on a fixed set of discrete grid points G. The

Algorithm 1 Prefix Tree Dictionary Search 1 n (n=1)o 1: INITIALIZE: C = Ci ; i ∈ [1,Ninit] 2: while n < Ns do n n ◦ 3: LIST all vacant Gridpoints Vi = TG(Ci ); i ∈ [1, |C |] Fig. 4. Expanded Beampattern for ROI (θmax < 30 ) 4: APPEND subarray at vacancies Vi, |Cn| Cˆ = S Cn × V of configurations depends on the allowed form factor, and the i i i=1 size of the subarray module, and an exhaustive search over all 5: PRUNE: Cn+1 ← Prune Cˆ configurations is computationally infeasible: for example, the 6: n = n + 1 number of configurations for a discrete grid of size 20 × 20 is 7: 20 end while 10 N of the order . We therefore propose a two-stage approach, 8: Return C s first performing a combinatorial search on a reduced search space, and then obtaining the final solution by searching over perturbations around the solution from the first stage. algorithm is described in Algorithm 1. We briefly discuss the key steps below.

A. Combinatorial search • INITIALIZE: In order to allow for sufficient exploration, 1 We reduce the solution space by removing geometrically we employ multiple random initializations Ci of the root “similar” arrays. We employ the covariance of the element po- node being placed on Ninit different locations on the grid. sitions, and pairwise element separations, as measures of sim- (For example, circular configurations cannot be obtained if the root subarray is fixed at the center.) ilarity. The choice of covariance of element positions Σ(C) n |C | |Vi| as similarity metric is motivated by its inverse proportionality • LIST: Define the operator TG : G i → G which n n to Cramer´ Rao Bound on accuracy of DoA estimation (see maps the set of subarrays centers Ci ∈ C to the set Section IV-A1). However, array configurations with similar of |Vi| vacant gridpoints in G available for placement array covariance but diverse beam attributes also exist: Figure of next subarray which are not blocked by the subarrays n 5 shows an example of two array configurations with different already placed at Ci . This operator also accounts for shapes but the same covariance. The arrays have similar additional surface area occupied by the subarray module beamwidth but their MSLL levels are different. We observe that apart from the physical antenna elements (see Appendix the variance of pairwise element distances ψ(C) is different C for details). th for these arrays, and use it as an indicator for these large • APPEND: The (n + 1) subarray configuration is con- n scale deviations. This allows us to reduce the dimensionality structed from the vacancies, Ci × Vi where × denotes cartesian product of sets. A temporary dictionary Cˆ is formed by inserting |V | = κ child nodes for each node C i 1 array th C in the n layer. 5 2 array C1 covariance • PRUNE: Nodes corresponding to “similar” configurations C 2 covariance are deleted based on the array shape parameters 0 ψ(C1)=6.6 1) Find eigenvalues (λ1, λ2) of the subarray center ψ(C2)=5.2 covariance matrix, Σ(C) and variance of array sep- -5 MSLL 2 1=-5.2 dB arations, ψ(C) = E (lij − E[lij]) , where, lij de- MSLL th th 2=-1.6 dB notes the distances betweeen i and j elements. -10 -5 0 5 10 2) Enumerate unique configurations by binning the (λ1, λ2, ψ) triplets over a 3-D grid with resolution τ Fig. 5. Super-arrays with equal covariance but different beam attributes. and randomly picking one configuration from each bin (see Appendix D for criteria to choose τ). of the solution space from 2 × Nsub down to 3 array shape The procedure is repeated until dictionary atoms reach the parameters: the eigenvalues (λ1, λ2) of Σ(C) and ψ(C). 1) Subarray Placement Algorithm: We construct a prefix desired number of subarrays i.e n = NS. All arrays in the tree dictionary to find feasible solutions using a breadth first dictionary C obtained from this algorithm satisfy the AOS(C) search based enumeration technique. The element position co- constraint by construction. This simplifies the constrained variance for an array of subarray can be uniquely represented Multi-objective optimization in Eq. (5) to by the covariance of its subarray centers, ΣD = ΣC + ΣD . e ∗ C = argminf(C, 1N ) Hence the super-array center covariance can be used instead of C∈C (8) IEEE TRANSACTIONS ON SIGNAL PROCESSING 6

B. Iterative Placement Refinement unknown deterministic constants. The joint probability density of received signal conditioned on (Θ, {α}K ) is given by, In the second stage, we try to improve the cost function (8) j=1 by applying small local perturbations (within a bin of the grid 2 ! Y 1 kx − αjs(uj)k p(x|Θ, α) = exp − (10) G) to the subarray positions obtained from the combinatorial πN σ2 σ2 search in the first stage, as described in Algorithm 2. uj ∈Θ For any DoA estimator Θˆ , the covariance of estimation error Algorithm 2 Local Refinements is defined as, init 1: INITIALIZE: C = C ; B = oversampled bin. " K # 2: while n < N do ˆ X T ref R(Θ) = E (u − uî)(u − uî) 3: for i = 1 to Ns do i=1 4: LIST Find positions available for adjustment, V = i R can be geometrically interpreted by its trace ptr(R ) T (C \ C ) B i which represents the expected overall Root mean square error 5: CORRECT: Select position with least cost C ← i (RMSE) in DoA estimation (see Appendix A). We use this min f ({(C \ C ), b} , 1) b∈Vi i measure to compare the performance of array designs in 6: end for Section V. For single source case (K = 1), the joint maximum 7: n = n + 1 likelihood estimator of u and α yields a noncoherent estimator 8: end while for u as follows: 9: Return C H 2 uˆML = arg max s(u) x (11) u For this case, we derive the Cramer Rao (CRB) and Ziv-Zakai Weighted Cost MSLL BeamWidth 5.4 (ZZB) bounds on R to assess the best possible estimation 0.02 accuracy of different designs. Although derived for single 0.018 -10.1 5.2 0.016 source case, we use these bounds for multiple source case -10.2 5 0.014 as well to compare DoA estimation performance. -10.3 0.012 4.8 1) Cramer´ Rao Bound : The Bayesian Cramer´ Rao Bound for this signal model is given by [20]: 2 4 6 8 2 4 6 8 2 4 6 8 −1 Iter. 1 Iter. 2 Iter. 3 Iter. 4 Iter. 5 CRB(R) = (JF + JP ) (12) where J ,J denote the Fisher Information Matrix (FIM) Fig. 6. Objective costs variation over iterative refinements. F P contributions from the observation and the prior distribution Figure 6 shows a sample of how costs are minimized using of DoA respectively. sequential refinement. After running few iterations, the final ∂2l(x|u) ∂2l(u) ◦ (JF )ij = −Ex,u , (JP )ij = −Eu array has 0.8 lower beamwidth, while keeping other beam ∂ui∂uj ∂ui∂uj attributes relatively unchanged. where l(x|u), l(u) are the conditional log likelihood and prior log likelihoods respectively. In addition the following IV. ESTIMATION-THEORETIC BENCHMARKS regularity condition needs to be satisfied, ∂l(x|u) We now seek to evaluate the efficacy of our sparse de- Ex,u = 0 signs for the canonical application of 2D DoA estimation. ∂u " N # We compare different array designs in terms of estimation- T T X jku di ∗ −jku di theoretic bounds as well as simulated performance using a Ex,u jk di xie − xi e = 0 super-resolution algorithm. For clarity in exposition, from here i=1 N N ! N onwards we overload u , [u, v] to denote the DoA. X ∗ X ∗ X jk α di − α di = jk(α − α ) di = 0 i=1 i=1 i=1 A. Signal Model In order to always satisfy this condition, we enforce the array PN element positions to be centered i.e. i=1 di = 0 We model the received signal from K sources in the scene For single source, the FIM is given by, with distinct DoAs Θ = [u1, u2, ···, uk] as " # 1 ∂s(u) H ∂s(u) K JF = − E (13) X σ2 ∂u ∂u x = αjs(uj) + z (9) 2 T j=1 = 2k γD D (14)

h T T iT which depends only on the element positions, D and Signal jku d1 jku dN where s(uj ) = e j , . . . , e j is the array re- to Noise ratio (SNR) (γ = |α|2/σ2). Assuming the DoA prior T ◦ sponse, z = [z1, . . . , zN ] is complex white noise such that to be uniformly distributed in the ROI (θ ≤ 30 ), the prior H 2 K E(zz ) = σ IN , and {α}j=1 are complex gains which are FIM simpliﬁes to JP = 1.343I2. IEEE TRANSACTIONS ON SIGNAL PROCESSING 7

2) Ziv-Zakai Bound: The CRB is a local bound, which maximized only when R(δ) is maximized. Therefore, for each accounts for estimation performance dependent on mainbeam, values of h we compute the maxδ:aT δ=h |R(δ)| numerically hence it is only useful at high SNR. In order to better by search over a discrete set of points on the line segment characterize the estimation performance of Sparse arrays at aT δ = h and substitue in (16). low SNR, we calculate the Ziv-Zakai Bound (ZZB) which incorporates the effect of sidelobes and predicts the threshold behavior. For any directional vector a = [cos ξ, sin ξ]T , the B. DoA estimation algorithm ZZB is given by [21] Grid-based sparse estimation for a set of DoAs models the Z ∞ Z the received signal (9) as follows: T a Ra ≥ V max A(u, δ)Pe(u, δ)du hdh T 0 δ:a δ=h x = S(Ψ)b + z (17) where, A(u, δ) = min {p(u), p(u + δ)}, V(.) is the valley where S(Ψ) = s(u ) ··· s(u ) contains the array re- filling function and Pe(u, δ) is error probability of the fol- 1 |Ψ| lowing vector parameter binary detection problem, sponse at discretized set of DoAs ui ∈ Ψ as columns. The nonzero entries in b point to presence of target in the 1 K H0 : uˆ = u; P r(H0) = , x ∼ p(x|u) corresponding DoA in Ψ. The DoA and gain pair (uî, αî)i=1 2 can be estimated by jointly minimizing the residual power, 1 H : uˆ = u + δ; P r(H ) = , x ∼ p(x|u + δ) 1 1 2 2 K X This error probability can be lower bounded by the minimum T (uˆ, αˆ) = x − αˆjs(uˆj) probability of error of the following optimal non-coherent j=1 detector: ( The NOMP algorithm summarized below provides a two stage u if ρ > ρ Decide(u) = 1 2 estimator: u + δ if ρ1 < ρ2 1) Detection: Using precomputed S(Ψ), coarse estimates H H of DoA and complex gain are obtained where ρ1 = |x s(u)|, ρ2 = |x s(u + δ)|. Given u = u0, 2 ρ1, ρ2 are rician distributed with scale parameter s = σ /M uˆ = arg max |s(u)H x|2 and non-centrality parameter ν = |α|N, |αR(δ)|N respec- u∈Ψ H tively where R(δ) = R(δx, δy) is the beampattern from (3). αˆ = s(u) x/N The error probability is given by [22] 2) Refinement: The estimates are refined using the Newton 1 P (u, δ) = (P r (ρ < ρ |u) + P r (ρ > ρ |u + δ)) method: nc 2 1 2 1 2 0 −1 = P r (ρ1 < ρ2|u) uˆ = uˆ − (H∇T (uˆ, αˆ)) ∇T (uˆ, αˆ) (18) 2 2 1 − a +b 0 0 H = Q (a, b) − e 2 I (ab) (15) αˆ = s(uˆ ) x/N (19) 1 2 0 where, where H∇T and ∇T denote the Hessian and gradient r γN of T (u, α) with respect to u at current estimate (uˆ, αˆ) a = 1 − p1 − |R(δ)|2 2 (see [23] for details). r r = γN p The algorithm is repeated with the residual signal, b = 1 + 1 − |R(δ)|2 0 0 2 x − αˆ s(û ) to estimate other DoAs. The refinement steps are repeated after each new detection for all DoAs in a cyclic which is not a function of u. For ROI in our case, the (max) manner for few rounds to improve accuracy. maximum error h(max) = aT δ = 1. Note that for The algorithm yields Kest DoA estimates, with estimation a uniformly distributed DoA in spherical coordinates (θ, φ), performance degrading when Kest does not match the true the distribution of u is not uniform. However for simplicity number of DoAs, KDoA. Hence, in order to evaluate the arrays of analysis, we make the assumption that u is uniformly independent of such errors, we implement both algorithm distributed on a circular disc. Hence, the ZZB expression where K = KDoA is known. simplifies to We also run extensive simulations with another state of Z 1 Z T the art algorithm, BPDN [14], with default parameters and 5 a Ra ≥ V max A(u)duPnc(δ) hdh refinement stages. The computational complexity of BPDN is 0 δ:aT δ=h Z 1 significantly higher than that of NOMP. The grid Ψ needs to be T ZZB(a Ra) = V max Pnc(δ) hdh (16) adapted at each iteration for BPDN, depending on the DoAs, T 0 δ:a δ=h whereas it remains fixed for NOMP S(Ψ), and can be pre- The maximum error probability over all directions δ cannot computed, which makes it suitable for faster implementation in be expressed as closed form expression. However, due to large arrays. Since the NOMP algorithm also yields somewhat the monotonicity of Marcum’s Q function, Q1(.) and Bessel better estimation accuracy than BPDN, we only present results th function of 0 order, I0(.), the error probability in (15) is obtained with NOMP here. IEEE TRANSACTIONS ON SIGNAL PROCESSING 8

Fig. 7. Beam patterns (Bottom row) for Designed (Left half) & benchmarking (right half) arrays.

V. NUMERICAL RESULTS TABLE I SAMPLE ARRAY CONFIGURATIONS A. Design of arrays Using the combinatorial search algorithm, we create a Shape α β γ MSLL BW ecc search space of array configurations, C ∈ C of size |C| = A1 1 0.1 0.1 -8 dB 5.7◦ 0 A2 8◦ 657, 000 for Nsub = 8 subarrays. We explore this search space 1 1 1 -10.2 dB 0 to obtain arrays using the placement optimization with various Compact - - - -12.8 dB 12.1◦ 0.7 choices of weights given to the cost function in Eq. (8). The Naive - - - -7.6 dB 7.9◦ 0.0 following key observations were made about the effect of these weights on subarray placement: Figure 7 shows the array designs obtained using our opti- • When primary weight is given towards minimizing mization approach and their beam patterns. The A2 array has beamwidth (larger α), the subarrays are widely distributed a sharp beamwidth and only 2.6 dB worse MSLL compared over the available aperture area. to the compact array. On the other hand, a naive sparse array • When we increase β to suppress MSLL, the array be- with circular arrangement of subarrays yields 2.6 dB higher comes restricted to smaller area. MSLL compared to A2 for similar beamwidth. Our designs A1, • The weight γ controls both the shape of main lobe as A2 exhibit several small sidelobes (the highest sidelobe for well as the position of sidelobes. A2 is -10.2 dB), whereas the naive array exhibits fewer but Based on these observations, we set the weights to obtain more pronounced sidelobes. Since large sidelobes and grating following sample array configurations: lobes can cause large errors in DoA estimation, we expect our 1) A1: Primary emphasis is given towards minimizing designs to yield better estimation performance, which is borne beamwidth by setting α = 1. If multiple arrays are out by the results presented in the next section. found near the minimum beamwidth, we give secondary priority to other attributes by giving small weight to B. Comparison of Estimation Performance them (β, γ = 0.1). 2) A2: In this case, we emphasize all beam attributes by We evaluate the arrays based on their DoA estimation setting all weights equal to 1. accuracy at different SNRs for both single and multiple source We compare these array designs against two simple array cases. We use the RMSE in estimating DoA for comparison p 2 p configurations, 1)“compact” array where subarrays are placed which is given by ¯ = E[|uˆ − u0)| ] = tr(R)/2. 1) Estimation bounds: The Cramer´ Rao Bound is evaluated together such that overall element pattern becomes a uniform p rectangular array, 2) “naive” array where subarrays are spread using (12), CRB(¯) = tr(CRB(R))/2 . The Ziv Zakai along a diamond shape such that its resultant beamwidth is Bound is evaluated using (16), equal to that of A2. Table I lists the weights and resulting q ZZB(¯) = ZZB(aT R a ) + ZZB(aT R a ) beam attributes of these arrays. 1 1 2 2 IEEE TRANSACTIONS ON SIGNAL PROCESSING 9

Overall RMSE uniform arrays, the minimum separation is typically deﬁned with respect to the DFT bin size (e.g. ∆DFT = 2π/L for an L- MLE A1 -5 element linear array). Since this quantity cannot be deﬁned for A2 non-uniform planar arrays, we choose a minimum separation Compact Naive halfway between the RMS beamwidths of the “compact” and sparse arrays, to capture the effect of both local errors and far -10 ZZB ambiguity errors due to sidelobes. In addition to RMSE vs SNR curves, we also analyze the

CRB distribution of error magnitudes. The complementary cumu- Error-15 (dB) lative distribution (CCDF) of the estimation errors is used to compare the “outage probability” corresponding to too large an error, which captures the impact of large sidelobes. -20 • With multiple sources, the estimation accuracy is de- graded by interference from other sources, and RMSE -13 -12 -11 -10 -9 -8 -7 -6 -5 does not converge to the single-target CRB. Fig. 9 shows SNR(dB) the estimation performance for strong and weak interference. Fig. 8. Comparison of estimation theoretic bounds for arrays. Interferers at 0 dB -5 where a1, a2 denote the directions of maximum and minimum A1 beamwidths of the array. We also computed the ML estimation -10 A2 RMSE Compact (MLE) error by Monte Carlo simulation using (11) with an Naive overcomplete dictionary of array responses. Figure 8 shows -15 the CRB, ZZB and MLE curves for all the arrays. CRB is proportional to beamwidth (see Appendix B for details). Error (dB) CRB -20 The ZZB bound converges to CRB at the so-called “ZZB threshold” SNR: when the SNR is below this threshold, far- ambiguities in DoA estimation caused by large sidelobes -25 dominate the MSE. The tradeoff between beamwidth and -15 -10 -5 0 5 MSLL is thus expected to translate to one between CRB SNR (dB) (better with smaller beamwidth) and ZZB threshold (worse Interferers at -6 dB with larger MSLL). Thus, as expected, “A1” array achieves -5 the lowest CRB, followed by “naive” and “A2” with equal A1 CRB, while “Compact” array has the largest beamwidth and -10 A2 hence highest CRB. The trend in MSLL is weakly reflected in RMSE Compact the ZZB thresholds (for a single target, sidelobes do limited Naive damage): the degradation in ZZB threshold, relative to that of -15 the compact array for the optimized arrays (A1, A2) is less than Error (dB) 1 dB, while the gain in CRB due to smaller beamwidth is 4 dB -20 CRB and 2 dB, respectively. The MLE error curve also agrees with the threshold behavior predicted by ZZB. We see in the next -25 set of results, however, that the size of the sidelobes becomes much more important when we consider multiple targets. -15 -10 -5 0 5 SNR (dB) 2) Estimation algorithm performance: We obtain DoA estimates using the NOMP algorithm [15], [23] with a known Fig. 9. Estimation accuracy with multiple targets. number of sources to compare the best case performance of these arrays. (The NOMP algorithm also performs as well as 1) Weak interference: When the interfering sources are the brute force MLE for a single target discussed earlier– 6 dB weaker than primary target, sparse arrays offer results omitted here.) The RMSE is evaluated across N = more than 5 dB SNR gain compared to compact 1024 DoAs uniformly sampled over the ROI (spherical cap arrays for SNR > −5 dB. Also, the difference of half angle 30◦). For evaluating the estimation performance between A2, naive array widens to about 1 dB in presence of multiple targets, (K = 5) we compute the in the threshold region indicating the benefit of RMS error in the DoA estimate for a primary target fixed at suppressing sidelobes. broadside, while interfering targets are distributed uniformly 2) Strong interference: When interfering sources have in ROI at separation of ∆u ≥ 0.16 or ∆θ ≥ 9.2◦ away from same magnitude, RMSE severely degrades for both primary target. This separation is imposed because the esti- arrays with high sidelobes (A1, Naive) as well as mation problem is ill-posed for DoAs in close proximity. For arrays with high beamwidth (Compact). On the IEEE TRANSACTIONS ON SIGNAL PROCESSING 10

other hand, A2 has lowest RMSE at SNR > −5 where x is the full measurement from (17) and Φ =

dB because of the dual benefit of small beamwidth diag(Φ1, ··· , ΦNs ) is the M × N measurement matrix con- and lower sidelobes. sisting of the subarrays measurement matrices as its block diagonals. Each subarray takes Mi compressive measurements with an independent Φ ∈ Mi×Ne whoose elements CCDF: Multiple Interferers -6 dB, SNR=-5.0 dB i C 100 are chosen uniformly and independently from QPSK sam- 1 ples √ {±1, ±j}. In addition, columns of Φi have unit Mi 2 norm to preserve signal norm on average (E ||ΦS(u)|| = ||S(u)||2) while scaling noise variance by N/M. The under- -2 10 A1 lying DoA, u can be extracted by minimizing the ML cost CCDF A2 function: Compact 2 K Naive X -4 y − Φ αˆjs(uˆj) 10 -30 -25 -20 -15 -10 -5 j=1 RMSE (dB) The efficacy of compressive parameter estimation in AWGN depends on preserving the geometric structure of the CCDF: Multiple interferers 0 dB, SNR=-5.0 dB 100 parametrized signals [17]. Specifically, if Φ satisfies the 2K isometry property for discretized basis S(Ψ) [17], |ΦS(Ψ)b|2 C(1 − ) ≤ ≤ C(1 + ) (20) |S(Ψ)b|2 10-2 A1

CCDF where C is a constant for any arbitrarily chosen 2K sparse A2 Compact vector b, the performance of the compressive system follows Naive that for the original system, except for an SNR penalty of 10-4 M/N. Figure 11 shows the minimum and maximum values of -30 -25 -20 -15 -10 -5 RMSE (dB) 6

4 Fig. 10. CCDF of estimation errors in multiple targets. 2 • The increase in estimation errors at high SNR is attributed 0 to ambiguity errors from sidelobes, hence the overall sidelobe suppression for the arrays can be compared -2 using the distribution of these error magnitudes. Fig dB scale -4 10 shows the CCDF curves of all arrays at SNR=−5 dB. The initial curvature of these curves (RMSE upto -6 -22 dB) is expected to depend on local errors, hence -8 the rate of change follows same order as CRB which 10 20 30 40 50 60 is A1 > A2 = naive > compact. But the curvature M reverses order at higher RMSE indicating the tradeoff with far-errors. We can see that A2 achieves lower outage Fig. 11. Maximum and minimum values of ratio in (20) for sparse array.

probability in both scenarios (e.g. for RMSE threshold set 6 to −15 dB) as it strikes a balance between near and far this ratio over 10 random realization of 8 sparse b for sparse errors. In contrast, both A1 and naive exhibit high outage array. The ratio is within [−5, 3] dB for M > 32 signifying probability due to frequent far ambiguity errors caused that 32 compressive measurements are sufﬁcient to estimate by higher sidelobes. K = 4 DoAs. Figure 12 shows estimation performance with M = 32 Therefore, depending on the expected magnitude of interferers compressive measurements collected across eight subarrays either one of the designed arrays with suitable sidelobe sup- (M = 4, i ∈ {1..8}). Comparing with Fig 8, we observe that pression can be selected. For a desired beamwidth reduction i the estimation algorithms preserve the same characteristics as our design algorithm yields an array superior to a naively with full measurements with approximately 6 dB SNR penalty designed sparse array. as expected (N/M = 4).

VI.COMPRESSIVE ESTIMATION VII.CONCLUSIONS We now evaluate the arrays for sparse estimation using Our results demonstrate that trading off beam width ver- compressive measurements at each subarray given by: sus side lobes when synthesizing a large effective aperture does indeed produce performance gains in bearing estima- y = Φx tion. Compared to a compact placement of subarrays, an IEEE TRANSACTIONS ON SIGNAL PROCESSING 11

Compressive Estimation R0(u) R(u). By taking the derivatives of (3), theTaylor 0 , series expanision up to second order is obtained as A1 k2 -5 A2 R(u) ≈ R(0) − uT DT Du (21) Compact N Naive -10 We deﬁne Half Power Beam Contour (HPBC) as the closed contour around mainbeam with {u : R(u) = 0.5R(0)}, which -15 is approximated as an ellipse using (21) as follows: Error (dB) N uT DT Du = R(0) (22) -20 2k2 Consider the eigendecomposition of DT D given by,

-25 T T T -6 -4 -2 0 2 4 D D = λ1p1p1 + λ2p2p2 (λ2 ≥ λ1) SNR (dB) The eigenvectors p2, p1 correspond to major and minor axis Fig. 12. Estimation performance with Compressive measurements. of HPBC ellipse respectively, and depend only on the element positions. optimized sparse placement produces smaller mean squared uˇ1 error because of its smaller beam width. Compared to a naive sparse placement, the control of side lobes via our optimized y placement produces the most significant gains when estimating the bearing for multiple sources. There are a number uˇo of interesting directions for further exploration. It may be ' z possible to improve our optimization framework by alternative approaches for pruning solutions, as well as applying generic optimization techniques (e.g., genetic optimization) initialized by our solutions. Exploring the application of our framework x for communications, where transmit and receive beamforming gains are fixed by the number of elements, but control of Fig. 13. 2D Beamwidth beam width and sidelobes affects interference, is an interesting direction. In the context of sensing, our work may be viewed Figure 13 shows the mainlobe of a beam and the dotted 1 2 as design of an individual sensor which can be placed within a shaded region represents its HPBC ellipse. uˇo, uˇ , uˇ corre- more comprehensive architecture, such as a network of sensors spond to unit vectors in the direction of main√ beam, vertex for localization and tracking. and co-vertex of the HPBC where, uˇ = [u, v, 1 − u2 − v2] is the unit vector towards directional cosine u = [u, v]. These APPENDIX A can be expressed as,       MEANSQUAREERRORIN 2D DOA ESTIMATION 0 sin ϑ cos φmax sin ϕ cos φmin For 2D DoA estimation, the error along any given angle ξ is uˇo = 0 , uˇ1 = sin ϑ sin φmax  , uˇ2 = sin ϕ sin φmin  T T given by a Ra, where a = [cos ξ, sin ξ] is the directional 1 cos ϑ cos ϕ cosine and R is the error covariance matrix. Assuming that (23) {νi, qi, i = 1, 2}, denote the eigenvalues and eigenvectors of where φmax, φmin are perpendicular azimuthal angles and ϑ, ϕ R, the MSE averaged over a (assume ξ uniform over [0, 2π]) are the maximum and minimum beamwidth angles subtended is given by from the major and minor axis of this ellipse to the mainbeam. h 2i h 2i T T T MSE = Ea a Ra = ν1Ea a q1 + ν2Ea a q2 360 −1 BWmax = ϑ = cos (uˇouˇ1) (24) 2 2 π ||q1|| ||q2|| 1 = ν1 + ν2 = (ν1 + ν2)/2 = tr(R) 2 2 2 360 −1 BWmin = ϕ = cos (uˇo.uˇ2) (25) 2 2 1 π where we have used E[cos ξ] = E[sin ξ] = 2 . Substituting the major and minor axis from (23) in (22), we APPENDIX B obtain 2D BEAMWIDTH &CRB 2 N p λ1 sin ϑ = R(0) =⇒ sin ϑ ≈ BWmax ∝ 1/ λ1 We define 2D beamwidth using the Taylor series expansion 2k2 of beampattern R (u) around mainlobe R (0). Since the 2 N p uo uo λ2 sin ϕ = R(0) =⇒ sin ϕ ≈ BWmin ∝ 1/ λ2 beampattern around the main lobe. and hence the beamwidth, 2k2 is invariant to beamforming direction (see II-C), we assume That is, the beamwidths along extremal directions are inversely T uo = 0 without loss of generality, and drop the subscript: proportional to the square roots of the eigenvalues of D D. IEEE TRANSACTIONS ON SIGNAL PROCESSING 12

Relation to CRB: Using (13), the error covariance matrix where is lower bounded by 2 1 υx υxυy 1 υxdxi υxdyi G = 2 , H = −1 N T −1 N υxυy υy N υydxi υydyi R ≥ CRB = JF = 2 D D 2k γ Using Weyl’s inequality [24] for real symmetric matrices, N 1 1 = p pT + p pT the eigenvalue perturbation is bounded as 2k2γ λ 1 1 λ 2 2 1 2 ¯ (using (λ2 ≥ λ1)) |λi − λi| ≤ kG + 2Hk2 ≤ kGk2 + 2 kHk2 Using Appendix A, the MSE can be lowerbounded by ≤ kGkF + 2 kHkF 1 N 1 1 The frobenius norms of G, H are MSE ≥ CRB = tr(J −1) = + 2 F 4k2γ λ λ q 1 2 kGk = υ2 + υ2/N N F x y = sin2(ϑ) + sin2(ϕ) q 4k2γ 2 2 kHkF = 2Ri υx + υy N (BWDoA)2 ∝ q 2 2 th SNR where Ri = dxi + dyi is the distance of the i element DoA p 2 2 p 2 2 from the array center. Hence, the overall variation of eigen- where BW = BWmax + BWmin = ϑ + ϕ is de- q 2 2 th fined as MSE beamwidth (sin θ ≈ θ for small angles θ). values with variation ∆e = υxi + υyi of the i element is APPENDIX C (2R + 1) |λ¯ − λ | ≤ i ∆ VACANCY SEARCH OPERATOR T i i N e Our reference subarray module shown in Fig. 14 occupies Thus, the eigenvalues are more sensitive to perturbations in the space in addition to antenna elements. In order to keep element locations of elements further from the center. For perturbations polarizations aligned, these modules can be placed in either up within one grid size used in our placement search algorithm, ◦ ◦ (0 ) or down (180 ) pose. We outline a procedure to list the the eigenvalue of the subarray center covariance can vary at n √ vacant gridpoints V = T(C ) where the new subarray can (2Ri+1) i i most by 2∆g, and we use this as a guideline for Nsub be placed without overlapping with already placed dormant discretizing the eigenvalues for.removing geometrically similar n subarrays at Ci . We define the subarray state as the center c configurations. of the element pattern and its pose ν, since vacancies depend on both parameters. ACKNOWLEDGMENT n c˜ = {c, ν}∀c ∈ Ci ,Vi This work was supported in part by Semiconductor Re- The pose variable ν ∈ {νu, νd, νf } denotes whether subarray search Corporation (SRC), DARPA, Center for Scientific Com- can be placed in up only(νu), down only(νd) or free pose (νf , puting at UCSB and National Science Foundation under grants either up or down) at the location c. For a given set of dormant CNS-1518812, CNS-1518632 and CNS-0960316. ñ ˜ subarray states, Ci we identify all vacant states Vi for placing the new subarray. Once a new subarray is placed, the states REFERENCES of all dormant subarrays are updated (e.g., a free pose may [1] A. Moffet, “Minimum-redundancy linear arrays,” IEEE Transactions on switch to an up pose if the down pose becomes infeasible). antennas and propagation, vol. 16, no. 2, pp. 172–175, 1968. [2] T. Birinci and Y. Tanık, “Optimization of nonuniform planar array geometry for direction of arrival estimation,” in Signal Processing Conference, 2005 13th European. IEEE, 2005, pp. 1–4. [3] V. Roy, S. P. Chepuri, and G. Leus, “Sparsity-enforcing sensor selection for doa estimation,” in Computational Advances in Multi-Sensor Adap- tive Processing (CAMSAP), 2013 IEEE 5th International Workshop on. IEEE, 2013, pp. 340–343. Fig. 14. The Subarray module and its two possible poses. Golden section are [4] A. Trucco, “Thinning and weighting of large planar arrays by simu- copper patch antennas on the green colored chip. lated annealing,” IEEE transactions on ultrasonics, ferroelectrics, and frequency control, vol. 46, no. 2, pp. 347–355, 1999. [5] P. Pal and P. P. Vaidyanathan, “Nested arrays in two dimensions, part i: Geometrical considerations,” IEEE Transactions on Signal Processing, APPENDIX D vol. 60, no. 9, pp. 4694–4705, Sept 2012. PERTURBATION OF ARRAY [6] E. Keto, “Hierarchical configurations for cross-correlation interferom- eters with many elements,” Journal of Astronomical Instrumentation, In order to design a bin size for pruning array configu- vol. 1, no. 01, p. 1250007, 2012. rations, we analyze the effect of perturbing the location of a [7] F. Athley, C. Engdahl, and P. Sunnergren, “On radar detection and direction finding using sparse arrays,” IEEE Transactions on Aerospace single array element on the eigenvalues of the array covariance and Electronic Systems, vol. 43, no. 4, 2007. matrix. Consider a small perturbation υ = [υx, υy] added to [8] R. Schmidt, “Multiple emitter location and signal parameter estimation,” th ¯ i array element position: di = di + υ. The covariance for IEEE transactions on antennas and propagation, vol. 34, no. 3, pp. 276– 280, 1986. the perturbed array is [9] R. Roy and T. Kailath, “Esprit-estimation of signal parameters via rota- T T T tional invariance techniques,” IEEE Transactions on acoustics, speech, ΣD¯ = D D + υ υ + 2di υ /N = ΣD + G + 2H and signal processing, vol. 37, no. 7, pp. 984–995, 1989. IEEE TRANSACTIONS ON SIGNAL PROCESSING 13

[10] K. T. Wong and M. D. Zoltowski, “Direction-finding with sparse Upamanyu Madhow received the bachelors de- rectangular dual-size spatial invariance array,” IEEE Transactions on gree in electrical engineering from IIT Kanpur in Aerospace and Electronic Systems, vol. 34, no. 4, pp. 1320–1336, 1998. 1985 and the Ph.D. degree in electrical engineering [11] M. D. Zoltowski and K. T. Wong, “Closed-form eigenstructure-based from the University of Illinois at UrbanaChampaign, direction finding using arbitrary but identical subarrays on a sparse Champaign, IL, USA, in 1990. He was a Research uniform cartesian array grid,” IEEE Transactions on Signal Processing, Scientist with Bell Communications Research, Mor- vol. 48, no. 8, pp. 2205–2210, 2000. ristown, NJ, USA. He was a Faculty Member with [12] M. Haardt and J. A. Nossek, “Simultaneous schur decomposition of the University of Illinois at UrbanaChampaign. He several nonsymmetric matrices to achieve automatic pairing in multidi- is currently a Professor of electrical and computer mensional harmonic retrieval problems,” IEEE Transactions on Signal engineering with the University of California at Processing, vol. 46, no. 1, pp. 161–169, 1998. Santa Barbara, Santa Barbara, CA, USA. He has [13] V. Vasylyshyn and O. Garkusha, “Direction finding using sparse array authored two textbooks, Fundamentals of Digital Communication (Cambridge composed of multiple identical subarrays,” in Antenna Theory and University Press, 2008) and Introduction to Communication Systems (Cam- Techniques, 2005. 5th International Conference on. IEEE, 2005, pp. bridge University Press, 2014). His current research interests focus on next- 273–276. generation communication, sensing and inference infrastructures centered [14] E. Van den Berg and M. P. Friedlander, “Sparse optimization with least- around millimeter-wave systems, and on robust machine learning. He was squares constraints,” SIAM Journal on Optimization, vol. 21, no. 4, pp. a recipient of the 1996 NSF CAREER Award and a co-recipient of the 2012 1201–1229, 2011. IEEE Marconi Prize Paper Award in Wireless Communications. He served [15] B. Mamandipoor, D. Ramasamy, and U. Madhow, “Newtonized orthog- as an Associate Editor for the IEEE Transactions on Communications, the onal matching pursuit: Frequency estimation over the continuum.” IEEE IEEE Transactions on Information Theory, and the IEEE Transactions on Trans. Signal Processing, vol. 64, no. 19, pp. 5066–5081, 2016. Information Forensics and Security. [16] C. Stoeckle, J. Munir, A. Mezghani, and J. A. Nossek, “Doa estimation performance and computational complexity of subspace-and compressed sensing-based methods,” in Smart Antennas (WSA 2015); Proceedings of the 19th International ITG Workshop on. VDE, 2015, pp. 1–6. Amin Arbabian (S06M12SM17) received the Ph.D. [17] D. Ramasamy, S. Venkateswaran, and U. Madhow, “Compressive pa- degree in electrical engineering and computer sci- rameter estimation in awgn.” IEEE Trans. Signal Processing, vol. 62, ence from the University of California at Berke- no. 8, pp. 2012–2027, 2014. ley (UC Berkeley), Berkeley, CA, USA, in 2011. [18] A. H. Nuttall and B. A. Cray, “Approximations to directivity for linear, From 2007 and to 2008, he was part of the Ini- planar, and volumetric apertures and arrays,” IEEE journal of oceanic tial Engineering Team, Tagarray, Inc., Palo Alto, engineering, vol. 26, no. 3, pp. 383–398, 2001. CA, USA. In 2010, he was with the Qualcomm’s [19] M. J. Lee, L. Song, S. Yoon, and S. R. Park, “Evaluation of directivity Corporate Research and Development Division, San for planar antenna arrays,” IEEE Antennas and Propagation Magazine, Diego, CA, USA, where he designed circuits for vol. 42, no. 3, pp. 64–67, 2000. next generation ultralow power wireless transceivers. [20] O. Lange and B. Yang, “Antenna geometry optimization for 2d direction- In 2012, he joined Stanford University, Stanford, of-arrival estimation for radar imaging,” in Smart Antennas (WSA), 2011 CA, USA, as an Assistant Professor of electrical engineering, where he is International ITG Workshop on. IEEE, 2011, pp. 1–8. currently a Frederick E. Terman Fellow with the School of Engineering. His [21] K. L. Bell, Y. Ephraim, and H. L. Van Trees, “Explicit ziv-zakai lower current research interests include high-frequency systems, medical imaging, bound for bearing estimation,” IEEE Transactions on Signal Processing, Internet-of Everything devices including wireless power delivery techniques, vol. 44, no. 11, pp. 2810–2824, 1996. and medical implants. Dr. Arbabian was a recipient or a co-recipient of [22] J. Salehi, “Digital communications 5e,” 2008. the 2015 NSF CAREER Award, the 2014 DARPA Young Faculty Award, [23] Z. Marzi, D. Ramasamy, and U. Madhow, “Compressive channel estima- the 2013 IEEE International Conference on Ultra-Wideband Best Paper tion and tracking for large arrays in mm-wave picocells,” IEEE Journal Award, the 2013 Hellman Faculty Scholarship, the 2010 IEEE Jack Kilby of Selected Topics in Signal Processing, vol. 10, no. 3, pp. 514–527, Award for Outstanding Student Paper of the International Solid-State Circuits April 2016. Conference, two-time Second Place Best Student Paper Award at the 2008 [24] R. A. Horn and C. R. Johnson, Matrix analysis. Cambridge university and 2011 Radio-Frequency Integrated Circuits (RFIC) Symposium, the 2009 press, 1990. Center for Information Technology Research in the Interest of Society at UC Berkeley Big Ideas Challenge Award and the UC Berkeley Bears Breaking Boundaries Award, and the 2010 2011 and 20142015 Qualcomm Innovation Fellowship. He currently serves on the TPC for the European Solid-State Circuits Conference and the RFIC Symposium.

Ali Sadri is Sr. Director of mmWave Standards and Advanced Technologies at Intel Corporation and the Chairman and CEO of the WiGig Alliance. Ali has 25 years of engineering, Scientiﬁc and Anant Gupta received the B.Tech. degree in elec- academic background in Wireless Communications tronics and electrical communication engineering system. His Professional work started at IBM in and the M.Tech. degree in telecommunication sys- year 1990 and a Visiting Professor at the Duke tems engineering from IIT Kharagpur in 2013. He University through year 2000. During years 2000- received the M.S. degree in electrical and computer 2002 he joined BOPS Inc., a startup company spe- engineering from the University of California at cialized in programmable DSP’s as the Sr. Director Santa Barbara (UCSB) in 2016. He is currently of the Communications and Advanced Development. working towards the Ph.D. degree in the Department In 2002 Ali joined Intel Corporation’s Mobile Wireless Division where he of Electrical and Computer Engineering, UCSB. His initiated and lead the standardization of the next generation High Throughput research interests include millimeter wave sensing, WLAN at Intel that became the IEEE 802.11n standards. Later Dr. Sadri statistical signal processing and machine learning. founded and lead the Wireless Gigabit Alliance in 2008 that created the ground breaking WiGig 60 GHz technology. In June 2013 WiGig Alliance merged with WiFi Alliance to advance and certify the WiGig programs within WiFi alliance framework. Currently Dr. Sadri is leading the mmWave advanced technology development that includes the next generation WiGig standards and mmWave technology for 5G cellular systems. Ali holds more than 100 Issued and pending patent applications in wired and wireless communications systems.